Understanding When First Supplying a Voltage Across an Inductor: No Momentary Voltage Spike

When discussing the behavior of inductors in electrical circuits, one often encounters the concept of voltage spikes. However, the situation changes dramatically when the voltage is first applied to an inductor. In this article, we will delve into the details of when a voltage is first applied across an inductor and explain why it does not produce a momentary voltage spike in the same manner as quick disconnection. Instead, the behavior can be understood through fundamental principles of inductance.

Inductance Basics

When a voltage is applied to an inductor, the inductor opposes changes in current due to its property of inductance. This means that when the voltage is first applied, the inductor initially resists the change in current, which affects its behavior and the resulting voltage behavior.

Current Build-Up

When a voltage is first supplied to the inductor, the current through the inductor starts at zero and increases gradually. This current build-up can be mathematically described by the following equation:

[ I_t frac{V}{R} left(1 - e^{-frac{R}{L}t}right) ]

where:

(V): The applied voltage (R): The resistance in the circuit (L): The inductance (t): Time

Voltage Across the Inductor

The voltage across the inductor at any moment is given by:

[ V_L L frac{dI}{dt} ]

At the moment the voltage is applied, (frac{dI}{dt}) is high as the current starts from zero. However, over time, as the current stabilizes, (frac{dI}{dt}) decreases. This means that the voltage across the inductor will initially be high but will stabilize as the current increases.

No Voltage Spike

Unlike the situation when the inductor is suddenly disconnected, which can cause a sudden change in current and lead to a high voltage spike due to the collapsing magnetic field, applying a voltage to the inductor does not create a spike. The inductor gradually allows current to build up, and the voltage across it decreases over time until it reaches a steady state.

Further Insights with Faraday's Law and Lenz's Law

To understand this behavior further, it is essential to consider Faraday's Law and Lenz's Law. According to Lenz's Law, the voltage induced in the inductor will always be in a direction that will create a magnetic flux to oppose the change in flux in the inductor. This law explains why the initiation of current in an inductor creates a significant back electromotive force (EMF) that opposes the rise of the current. Thus, the inductor acts as a current-limiting device, causing the current to increase much more slowly than it would without the inductor.

When switching the device off, the situation is reversed. The sudden drop in the large current results in a high drop in the flux, which produces a huge back EMF to try and maintain the flux. This illustrates why a momentary voltage spike can occur during the disconnection of the inductor.

However, in the case of initially applying voltage, the behavior remains under control, and the transient response is not manifested as a voltage spike.

Conclusion

In summary, while there is a transient response when voltage is first applied to an inductor, it does not manifest as a voltage spike in the same way as when the inductor is suddenly disconnected. The inductor gradually allows current to build up, with the voltage across it decreasing over time until it reaches a steady state. This behavior is fundamentally different from the transient conditions that can occur during disconnection, where a high voltage spike can be observed due to the collapsing magnetic field.

References

Faraday, M. (1831). Experimental researches in electricity. Part I. New Series, 1, 1-52. Lenz, H. (1834). Ueber das Zeitalter der Elektricit?t. Annalen der Physik und Chemie, 106(6), 262-266.

For further reading, explore the works of Michael Faraday and Heinrich Lenz to gain a deeper understanding of the principles governing inductance and transient behavior in electrical circuits.